Premium
Convolution product formula for associated homogeneous distributions on R
Author(s) -
Franssens Ghislain R.
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1397
Subject(s) - convolution (computer science) , mathematics , convolution power , product (mathematics) , associative property , homogeneous , domain (mathematical analysis) , set (abstract data type) , pure mathematics , discrete mathematics , algebra over a field , mathematical analysis , fourier transform , combinatorics , computer science , geometry , fourier analysis , artificial neural network , fractional fourier transform , programming language , machine learning
The set of Associated Homogeneous Distributions (AHDs) on R , ℋ′( R ), consists of distributional analogues of power‐log functions with domain in R . This set contains the majority of the (one‐dimensional) distributions typically encountered in physics applications. In earlier work of the author it was shown that ℋ′( R ) admits a closed convolution structure, provided that critical convolution products are defined by a functional extension process. In this paper, the general convolution product formula is derived. Convolution of AHDs on R is found to be associative, except for critical triple products. Critical products are shown to be non‐associative in a minimal and interesting way. Copyright © 2010 John Wiley & Sons, Ltd.